Data Representation in Computer Systems

Data Representation in Computer Systems
Chapter 2 Data Representation in Computer Systems

Chapter 2 Objectives Understand the fundamentals of numerical data representation and manipulation in digital computers Master the skill of converting between various radix systems Understand how errors can occur in computations because of overflow and truncation Understand the fundamental concepts of floating-point representation Gain familiarity with the most popular character codes. Understand the concepts of error detecting and correcting codes

Introduction In chapter 1, we briefly saw that computer technology is based on integrated circuits (IC), which themselves are based on transistors The idea is that ICs store circuits that operate on electrical current by either letting current pass through, or blocking the current So, we consider every circuit in the computer to be in a state of on or off (1 or 0) Because of this, the computer can store information in binary form But we want to store information from the real world, information that consists of numbers, names, sounds, pictures, instructions So we need to create a representation for these types of information using nothing but 0s and 1s That’s the topic for this chapter

Introduction A bit is the most basic unit of information in a computer
A byte is a group of eight bits, the smallest possible addressable unit of computer storage. The term, “addressable,” means that a particular byte can be retrieved according to its location in memory A word is a contiguous group of bytes. Word sizes of 16, 32, or 64 bits are most common. In a word-addressable system, a word is the smallest addressable unit of storage. A group of four bits is called a nibble (or nybble).

2.2 Positional Numbering Systems
Bytes store numbers using the position of each bit to represent a power of 2. The binary system is also called the base-2 system. Our decimal system is the base-10 system. It uses powers of 10 for each position in a number. Any integer quantity can be represented exactly using any base (radix).

The decimal number 947 in powers of 10 is: The decimal number in powers of 10 is: 9    10 0 5     10 0 + 4   10 -2

The binary number in powers of 2 is: When the radix of a number is something other than 10, the base is denoted by a subscript. Sometimes, the subscript 10 is added for emphasis: = 2510 1      2 0 = = 25

2.3 Decimal to Binary Conversions
Because binary numbers are the basis for all data representation in digital computer systems, it is important that you become proficient with this radix system. Your knowledge of the binary numbering system will enable you to understand the operation of all computer components as well as the design of ISAs.

In an earlier slide, we said that every integer value can be represented exactly using any radix system. You can use either of two methods for radix conversion: the subtraction method and the division remainder method. The subtraction method is more intuitive, but cumbersome. It does, however reinforce the ideas behind radix mathematics.

Using division to convert integers from decimal to some other radix.
It employs the idea that successive division by a base is equivalent to successive subtraction by powers of the base. Let’s use the division remainder method to again convert 190 in decimal to base 3.

Converting 190 to base 3... First we take the number that we wish to convert and divide it by the radix in which we want to express our result. In this case, 3 divides times, with a remainder of 1. Record the quotient and the remainder Continue in this way until the quotient is zero

Fractional values can be approximated in all base systems Unlike integer values, fractions do not necessarily have exact representations under all radices. The quantity ½ is exactly representable in the binary and decimal systems, but is not in the ternary (base 3) numbering system.

Fractional decimal values have nonzero digits to the right of the decimal point. Fractional values of other radix systems have nonzero digits to the right of the radix point. Numerals to the right of a radix point represent negative powers of the radix: = 4   10 -2 = 1   2 -2 = ½ ¼ = = 0.75

Using the multiplication method to convert the decimal to binary, we multiply by the radix 2. The first product carries into the units place. Ignoring the value in the units place at each step, continue multiplying each fractional part by the radix. You are finished when the product is zero, or until you have reached the desired number of binary places. Our result, reading from top to bottom is: =

Hexadecimal The binary numbering system is the most important radix system for digital computers However, it is difficult to read long strings of binary numbers For compactness and ease of reading, binary values are usually expressed using the hexadecimal The hexadecimal numbering system uses the numerals 0 through 9 and the letters A through F. The decimal number 26 is 1A16. Thus, to convert from binary to hexadecimal, all we need to do is group the binary digits into groups of four.

Using groups of hextets, the binary number (= ) in hexadecimal is: Octal (base 8) values are derived from binary by using groups of three bits (8 = 23): Octal was very useful when computers used six-bit words.

Signed Integer Representation
The conversions we have so far presented have involved only positive numbers. To represent negative values, computer systems allocate the high-order bit to indicate the sign of a value. The high-order bit is the leftmost bit in a byte. It is also called the MSB. The remaining bits contain the value of the number. When an integer variable is declared in a program, many programming languages automatically allocate a storage area that includes a sign as the first bit of the storage location. Three ways in which signed binary numbers may be expressed Signed magnitude One’s complement Two’s complemen

Signed Magnitude Representation
In an 8-bit word, signed magnitude representation places the absolute value of the number in the 7 bits to the right of the sign bit In 8-bit signed magnitude Positive 3 is Negative 3 is Computers perform arithmetic operations on signed magnitude numbers in much the same way as humans carry out pencil and paper arithmetic. Ignoring the signs of the operands while performing a calculation, applying the appropriate sign after the calculation is complete

Example one Using signed magnitude binary arithmetic, find the sum of 75 and 46. Solution convert 75 and 46 to binary, and arrange as a sum, but separate the (positive) sign bits from the magnitude bits. We were careful to pick two values whose sum would fit into 7 bits. If that is not the case, we have a problem

Example Two Using signed magnitude binary arithmetic, find the sum of 107 and 46. We see that the carry from the seventh bit overflows and is discarded, giving us the erroneous result = 25

Example Three Using signed magnitude binary arithmetic, find the sum of - 46 and - 25. Because the signs are the same, all we do is add the numbers and supply the negative sign when we are done

Example Four Using signed magnitude binary arithmetic, find the sum of 46 and - 25. Mixed sign addition (or subtraction) is done the same way The sign of the result gets the sign of the number that is larger. Note the “borrows” from the second and sixth bits.

Signed Magnitude Representation
Advantage easy for people to understand. Disadvantage of signed magnitude It allows 2 different representations for zero: positive zero and negative zero. It requires complicated computer hardware For these reasons (among others) computers systems employ Complement systems for numeric value representation.

Complement Systems In complement systems, negative values are represented by some difference between a number and its base. In diminished radix complement systems, a negative value is given by the difference between the absolute value of a number and one less than its base. In the binary system, this gives us one’s complement. It amounts to little more than flipping the bits of a binary number

One’s Complement With one’s complement addition, the carry bit is “carried around” and added to the sum. In 8-bit one’s complement positive 3 is: Negative 3 is: In one’s complement, as with signed magnitude, negative values are indicated by a 1 in the high order bit. Complement systems are useful because they eliminate the need for subtraction. The difference of two values is found by adding the minuend to the complement of the subtrahend.

One’s Complement Example One
Using one’s complement binary arithmetic, find the sum of 48 and - 19 We note that 19 in one’s complement -19 in one’s complement is

One’s Complement Although the “end carry around” adds some complexity, one’s complement is simpler to implement than signed magnitude Disadvantage two different representations for zero Two’s complement solves this problem. Two’s complement is the radix complement of the binary numbering system.

Two’s Complement Example One 19 in one’s complement is 00010011
To express a value in two’s complement: If the number is positive, just convert it to binary and you’re done. If the number is negative, find the one’s complement of the number and then add 1. With two’s complement arithmetic, all we do is add our two binary numbers. Just discard any carries emitting from the high order bit. Example One Using Two’s complement binary arithmetic, find the sum of 48 and - 19. 19 in one’s complement is -19 in one’s complement is -19 in two’s complement is

Overflow When we use any finite number of bits to represent a number, we always run the risk of the result of our calculations becoming too large to be stored in the computer. While we can’t always prevent overflow, we can always detect overflow. In complement arithmetic, an overflow condition is easy to detect.

Two’s Complement Example Two
Using two’s complement binary arithmetic, find the sum of 107 and 46. We see that the nonzero carry from the seventh bit overflows into the sign bit, giving us the erroneous result: = -103 Rule for detecting signed two’s complement overflow When the “carry in” and the “carry out” of the sign bit differ, overflow has occurred.

Do we need unsigned numbers?
Signed and unsigned numbers are both useful e.g, memory addresses are always unsigned Using the same number of bits, unsigned integers can express twice as many values as signed numbers.

2.5 Floating-Point Representation
The signed magnitude, one’s complement, and two’s complement representation that we have just presented deal with integer values only. Without modification, these formats are not useful in scientific or business applications that deal with real number values. Floating-point representation solves this problem.

Programmers can perform floating-point calculations using any integer format. This is called floating-point emulation, because floating point values aren’t stored as such, we just create programs that make it seem as if floating-point values are being used. Most of today’s computers are equipped with specialized hardware that performs floating-point arithmetic with no special programming required.

Floating-point numbers allow an arbitrary number of decimal places to the right of the decimal point. For example: 0.5  0.25 = 0.125 They are often expressed in scientific notation. For example: 0.125 = 1.25  10-1 5,000,000 = 5.0  106 Computers use a form of scientific notation for floating-point representation Numbers written in scientific notation have three components

Computer representation of a floating-point number consists of three fixed-size fields The one-bit sign field is the sign of the stored value The size of the exponent field, determines the range of values that can be represented The size of the significand determines the precision of the representation. The significand of a floating-point number is always preceded by an implied binary point. Thus, the significand always contains a fractional binary value. The exponent indicates the power of 2 to which the significand is raised

Example Express 3210 in the simplified 14-bit floating-point model.
We know that 32 is 25. So in (binary) scientific notation 32 = 1.0 x 25 = 0.1 x 26. Using this information, we put 110 (= 610) in the exponent field and 1 in the significand as shown.

Another problem with our system is that we have made no allowances for negative exponents. We have no way to express 0.5 (=2 -1)! (Notice that there is no sign in the exponent field!) All of these problems can be fixed with no changes to our basic model.

To resolve the problem of synonymous forms, we will establish a rule that the first digit of the significand must be 1. This results in a unique pattern for each floating-point number. In the IEEE-754 standard, this 1 is implied meaning that a 1 is assumed after the binary point. By using an implied 1, we increase the precision of the representation by a power of two. (Why?) In our simple instructional model, we will use no implied bits.

To provide for negative exponents, we will use a biased exponent. A bias is a number that is approximately midway in the range of values expressible by the exponent. We subtract the bias from the value in the exponent to determine its true value. In our case, we have a 5-bit exponent. We will use 16 for our bias. This is called excess-16 representation. In our model, exponent values less than 16 are negative, representing fractional numbers.

Example: Express 3210 in the revised 14-bit floating-point model. We know that 32 = 1.0 x 25 = 0.1 x 26. To use our excess 16 biased exponent, we add 16 to 6, giving 2210 (=101102). Graphically:

Example: Express in the revised 14-bit floating-point model. We know that is So in (binary) scientific notation = 1.0 x 2-4 = 0.1 x 2 -3. To use our excess 16 biased exponent, we add 16 to -3, giving 1310 (=011012).

Example Express -26.62510 in the revised 14-bit floating-point model.
We find = Normalizing, we have: = x 2 5. To use our excess 16 biased exponent, we add 16 to 5, giving 2110 (=101012). We also need a 1 in the sign bit.

The IEEE-754 single precision floating point standard uses bias of 127 over its 8-bit exponent. An exponent of 255 indicates a special value. If the significand is zero, the value is  infinity. If the significand is nonzero, the value is NaN, “not a number,” often used to flag an error condition. The double precision standard has a bias of 1023 over its 11-bit exponent. The “special” exponent value for a double precision number is 2047, instead of the 255 used by the single precision standard.

Both the 14-bit model that we have presented and the IEEE-754 floating point standard allow two representations for zero. Zero is indicated by all zeros in the exponent and the significand, but the sign bit can be either 0 or 1. This is why programmers should avoid testing a floating-point value for equality to zero. Negative zero does not equal positive zero.

Floating-point addition and subtraction are done using methods analogous to how we perform calculations using pencil and paper. The first thing that we do is express both operands in the same exponential power, then add the numbers, preserving the exponent in the sum. If the exponent requires adjustment, we do so at the end of the calculation.

Example: Find the sum of 1210 and using the 14-bit floating-point model. We find 1210 = x And = x 2 1 = x 2 4. Thus, our sum is x 2 4.

Floating-point multiplication is also carried out in a manner akin to how we perform multiplication using pencil and paper. We multiply the two operands and add their exponents. If the exponent requires adjustment, we do so at the end of the calculation.

Example: Find the product of 1210 and using the 14-bit floating-point model. We find 1210 = x And = x 2 1. Thus, our product is x 2 5 = x 2 4. The normalized product requires an exponent of 2010 =

No matter how many bits we use in a floating-point representation, our model must be finite. The real number system is, of course, infinite, so our models can give nothing more than an approximation of a real value. At some point, every model breaks down, introducing errors into our calculations. By using a greater number of bits in our model, we can reduce these errors, but we can never totally eliminate them.

Our job becomes one of reducing error, or at least being aware of the possible magnitude of error in our calculations. We must also be aware that errors can compound through repetitive arithmetic operations. For example, our 14-bit model cannot exactly represent the decimal value In binary, it is 9 bits wide: =

When we try to express in our 14-bit model, we lose the low-order bit, giving a relative error of: If we had a procedure that repetitively added 0.5 to 128.5, we would have an error of nearly 2% after only four iterations. 128.5  0.39%

Floating-point errors can be reduced when we use operands that are similar in magnitude. If we were repetitively adding 0.5 to 128.5, it would have been better to iteratively add 0.5 to itself and then add to this sum. In this example, the error was caused by loss of the low-order bit. Loss of the high-order bit is more problematic.

Floating-point overflow and underflow can cause programs to crash. Overflow occurs when there is no room to store the high-order bits resulting from a calculation. Underflow occurs when a value is too small to store, possibly resulting in division by zero. Experienced programmers know that it’s better for a program to crash than to have it produce incorrect, but plausible, results.

When discussing floating-point numbers, it is important to understand the terms range, precision, and accuracy. The range of a numeric integer format is the difference between the largest and smallest values that is can express. Accuracy refers to how closely a numeric representation approximates a true value. The precision of a number indicates how much information we have about a value

Most of the time, greater precision leads to better accuracy, but this is not always true. For example, is a value of pi that is accurate to two digits, but has 5 digits of precision. There are other problems with floating point numbers. Because of truncated bits, you cannot always assume that a particular floating point operation is commutative or distributive.

This means that we cannot assume: (a + b) + c = a + (b + c) or a*(b + c) = ab + ac Moreover, to test a floating point value for equality to some other number, first figure out how close one number can be to be considered equal. Call this value epsilon and use the statement: if (abs(x) < epsilon) then ...

2.6 Character Codes We have seen how digital computers use the binary system to represent and manipulate numeric values. We have yet to consider how these internal values can be converted to a form that is meaningful to humans. The manner in which this is done depends on both the coding system used by the computer and how the values are stored and retrieved

Binary-Coded Decimal BCD is a numeric coding system used primarily in IBM mainframe and midrange systems. As its name implies, BCD encodes each digit of a decimal number to a 4-bit binary form. When stored in an 8-bit byte, the upper nibble is called the zone and the lower part is called the digit. The high-order nibble in a BCD byte is used to hold the sign, which can have one of three values An unsigned number is indicated with 1111 a positive number is indicated with 1100 a negative number is indicated with 1101

EXAMPLE Represent 1265 in 3 bytes using packed BCD.
The zoned-decimal coding for 1265 is: After packing, this string becomes: Adding the sign after the low-order digit and padding the high-order digit with ones in 3 bytes we have

2.6 Character Codes As computers have evolved, character codes have evolved Larger computer memories and storage devices permit richer character codes The earliest computer coding systems used six bits. Binary-coded decimal (BCD) was one of these early codes. It was used by IBM mainframes in the 1950s and 1960s.

2.6 Character Codes In 1964, BCD was extended to an 8-bit code, Extended Binary-Coded Decimal Interchange Code (EBCDIC). EBCDIC was one of the first widely-used computer codes that supported upper and lowercase alphabetic characters, in addition to special characters, such as punctuation and control characters. EBCDIC and BCD are still in use by IBM mainframes today.

2.6 Character Codes Other computer manufacturers chose the 7-bit ASCII (American Standard Code for Information Interchange) as a replacement for 6-bit codes. While BCD and EBCDIC were based upon punched card codes, ASCII was based upon telecommunications (Telex) codes. Until recently, ASCII was the dominant character code outside the IBM mainframe world.

2.6 Character Codes Many of today’s systems embrace Unicode, a 16-bit system that can encode the characters of every language in the world. The Java programming language, and some operating systems now use Unicode as their default character code. The Unicode codespace is divided into six parts. The first part is for Western alphabet codes, including English, Greek, and Russian.

2.6 Character Codes The Unicode codespace allocation is shown at the right. The lowest-numbered Unicode characters comprise the ASCII code. The highest provide for user-defined codes.

2.8 Error Detection and Correction
It is physically impossible for any data recording or transmission medium to be 100% perfect 100% of the time over its entire expected useful life. As more bits are packed onto a square centimeter of disk storage, as communications transmission speeds increase, the likelihood of error increases-- sometimes geometrically. Thus, error detection and correction is critical to accurate data transmission, storage and retrieval.

Error Detection and Correction
Check digits, appended to the end of a long number can provide some protection against data input errors. The last character of UPC barcodes and ISBNs are check digits. Longer data streams require more economical and sophisticated error detection mechanisms. Cyclic redundancy checking (CRC) codes provide error detection for large blocks of data.

Checksums and CRCs are examples of systematic error detection. In systematic error detection a group of error control bits is appended to the end of the block of transmitted data. This group of bits is called a syndrome. CRCs are polynomials over the modulo 2 arithmetic field. The mathematical theory behind modulo 2 polynomials is beyond our scope. However, we can easily work with it without knowing its theoretical underpinnings.

Modulo 2 arithmetic works like clock arithmetic. In clock arithmetic, if we add 2 hours to 11:00, we get 1:00. In modulo 2 arithmetic if we add 1 to 1, we get 0. The addition rules couldn’t be simpler: 0 + 0 = = 1 1 + 0 = = 0 You will fully understand why modulo 2 arithmetic is so handy after you study digital circuits in Chapter 3.

Find the quotient and remainder when is divided by 1101 in modulo 2 arithmetic. As with traditional division, we note that the dividend is divisible once by the divisor. We place the divisor under the dividend and perform modulo 2 subtraction.

Find the quotient and remainder when is divided by 1101 in modulo 2 arithmetic… Now we bring down the next bit of the dividend. We see that is not divisible by So we place a zero in the quotient.

Find the quotient and remainder when is divided by 1101 in modulo 2 arithmetic… 1010 is divisible by 1101 in modulo 2. We perform the modulo 2 subtraction.

Find the quotient and remainder when is divided by 1101 in modulo 2 arithmetic… We find the quotient is 1011, and the remainder is 0010. This procedure is very useful to us in calculating CRC syndromes. Note: The divisor in this example corresponds to a modulo 2 polynomial: X 3 + X

Suppose we want to transmit the information string: The receiver and sender decide to use the (arbitrary) polynomial pattern, 1101. The information string is shifted left by one position less than the number of positions in the divisor. The remainder is found through modulo 2 division (at right) and added to the information string: =

If no bits are lost or corrupted, dividing the received information string by the agreed upon pattern will give a remainder of zero. We see this is so in the calculation at the right. Real applications use longer polynomials to cover larger information strings. Some of the standard poly-nomials are listed in the text.

Data transmission errors are easy to fix once an error is detected. Just ask the sender to transmit the data again. In computer memory and data storage, however, this cannot be done. Too often the only copy of something important is in memory or on disk. Thus, to provide data integrity over the long term, error correcting codes are required.

Hamming codes and Reed-Soloman codes are two important error correcting codes. Reed-Soloman codes are particularly useful in correcting burst errors that occur when a series of adjacent bits are damaged. Because CD-ROMs are easily scratched, they employ a type of Reed-Soloman error correction. Because the mathematics of Hamming codes is much simpler than Reed-Soloman, we discuss Hamming codes in detail.

Hamming codes are code words formed by adding redundant check bits, or parity bits, to a data word. The Hamming distance between two code words is the number of bits in which two code words differ. The minimum Hamming distance for a code is the smallest Hamming distance between all pairs of words in the code. This pair of bytes has a Hamming distance of 3:

Suppose we have a set of n-bit code words consisting of m data bits and r (redundant) parity bits. An error could occur in any of the n bits, so each code word can be associated with n erroneous words at a Hamming distance of 1. Therefore,we have n + 1 bit patterns for each code word: one valid code word, and n erroneous words.

With n-bit code words, we have 2 n possible code words consisting of 2 m data bits (where n = m + r). This gives us the inequality: (n + 1)  2 m  2 n Because n = m + r, we can rewrite the inequality as: (m + r + 1)  2 m  2 m + r or (m + r + 1)  2 r This inequality gives us a lower limit on the number of check bits that we need in our code words.

Suppose we have data words of length m = 4. Then: (4 + r + 1)  2 r implies that r must be greater than or equal to 3. This means to build a code with 4-bit data words that will correct single-bit errors, we must add 3 check bits. Finding the number of check bits is the hard part. The rest is easy.

Suppose we have data words of length m = 8. Then: (8 + r + 1)  2 r implies that r must be greater than or equal to 4. This means to build a code with 8-bit data words that will correct single-bit errors, we must add 4 check bits, creating code words of length 12. So how do we assign values to these check bits?

With code words of length 12, we observe that each of the digits, 1 though 12, can be expressed in powers of 2. Thus: 1 = = = 2 = = = 3 = = = 4 = = = 1 (= 20) contributes to all of the odd-numbered digits. 2 (= 21) contributes to the digits, 2, 3, 6, 7, 10, and 11. . . . And so forth . . . We can use this idea in the creation of our check bits.

Using our code words of length 12, number each bit position starting with 1 in the low-order bit. Each bit position corresponding to an even power of 2 will be occupied by a check bit. These check bits contain the parity of each bit position for which it participates in the sum.

Since 2 (= 21) contributes to the digits, 2, 3, 6, 7, 10, and 11. Position 2 will contain the parity for bits 3, 6, 7, 10, and 11. When we use even parity, this is the modulo 2 sum of the participating bit values. For the bit values shown, we have a parity value of 0 in the second bit position. What are the values for the other parity bits?

The completed code word is shown above. Bit 1checks the digits, 3, 5, 7, 9, and 11, so its value is 1. Bit 4 checks the digits, 5, 6, 7, and 12, so its value is 1. Bit 8 checks the digits, 9, 10, 11, and 12, so its value is also 1. Using the Hamming algorithm, we can not only detect single bit errors in this code word, but also correct them!

Suppose an error occurs in bit 5, as shown above. Our parity bit values are: Bit 1 checks digits, 3, 5, 7, 9, and 11. Its value is 1, but should be zero. Bit 2 checks digits 2, 3, 6, 7, 10, and 11. The zero is correct. Bit 4 checks digits, 5, 6, 7, and 12. Its value is 1, but should be zero. Bit 8 checks digits, 9, 10, 11, and 12. This bit is correct.

We have erroneous bits in positions 1 and 4. With two parity bits that don’t check, we know that the error is in the data, and not in a parity bit. Which data bits are in error? We find out by adding the bit positions of the erroneous bits. Simply, = 5. This tells us that the error is in bit 5. If we change bit 5 to a 1, all parity bits check and our data is restored.

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